Graduate Student Seminar, 4:30 pm May 2, 2005; HH127


Jacob Heidenreich


The Great Change: Mathematics at the Turn of the (Last) Century


In the late 19th and early 20th centuries, a great shift occured in mathematics. Mathematicians thought differently about what the goals and techniques of mathematics should be. Before the shift, research mathematics looked like a complex version of undergraduate Calculus (what's the deriviative of x^2 - x at the point...). After the shift, research mathematics looked like a complex version of graduate Real Analysis (let X be a measure space and S be the sigma algebra of measurable subsets...). Mathematics changed from being about something to being about nothing. Specifically, it changed from being about particular things (solve *this* equation) to being about nothing *in particular* (anything that satisfies a certain set of axioms). My talk will study this shift, the mathematical advances which motivated it, and the many arguments and name-callings it gave rise to ("That's not mathematics, that's theology!"-- Gordon). In particular, we'll look at the Hilbert-Frege debate over what mathematical axioms and definitions are. Frege argued that a statement is considered an axiom if its truth is beyond doubt (think of the Euclidean Axioms) and we should call a statement an axiom only if we're certain of it. Hilbert said an axiom set was really a definition of a kind of mathematical structures (think of the group theory axioms as defining what a group is) and that this is how a mathematician should think of *any* axiom set, including the Euclidean Axioms. Frege said Euclidean Geometry is "about" the Euclidean plane, and Hilbert said Euclidean Geometry is "about" any structure that satisfies the Euclidean axioms -- we must not say what. As Russell put it, "Mathematics is the subject in which we don't know what we are talking about, nor whether what we are saying is true."

To volunteer to give a talk, or for any other questions regarding this schedule, contact Sara Miller